If , describe the transformation from to if
If, alternatively, you are looking for a description of the graph of the new function (i.e. shifts/rotations about lines/stretching/skewing/etc.), then break it down step-by-step.
Compare the graph of to the graph of . Then compare those to the graph of . Next, compare them to the graph of . Finally, put it all together as you compare it to the graph of .
Let's consider what happens when we plug in numbers. If we plug in 5 to , we get . Going from the graph of to the graph of , the x-position of is going from x=4 to x=5. In other words, it is shifting to the right one unit. Now let's consider . Now, is at x=-4 in our original graph, but it is at x=5 in this graph. Does this mean it is shifting to the right nine units? Not quite. Let's look at some more data to see what's happening. is the y-value at x=-3 in our original graph, but it is the y-value at x=4 in . So now the shift is only 7 units. The negative sign is actually creating a reflection across some axis. The closer we get to , the less the points need to shift from the original function to our new function. In other words, this is a reflection across the line x=1. Next, is taking the value and taking its reflection across the x-axis. Finally, is taking the graph of and shifting it down seven units.